Functions and Sequences Flashcards

A function f satisfies f(x + y) = f(x) + f(y) for all real numbers x and y, and f(1) = 4. What is f(7)?

Answer: 28

By the Cauchy functional equation, f(n) = 4n for all integers, so f(7) = 4 × 7 = 28.

If f(2x) = f(x) + x for all x > 0 and f(1) = 0, what is f(8)?

Answer: 7

f(2) = f(1)+1 = 1, f(4) = f(2)+2 = 3, f(8) = f(4)+4 = 7.

Let f(x) = ax + b with a > 0. If f(f(x)) = 9x + 16, what is a + b?

Answer: 7

f(f(x)) = a²x + b(a+1) = 9x + 16, so a² = 9 → a = 3, then 4b = 16 → b = 4, giving a + b = 7.

How many functions f: {1, 2, 3} → {1, 2, 3} satisfy f(f(x)) = x for all x in the domain?

Answer: 4

Such involutions are: the identity, swap(1,2) fix 3, swap(1,3) fix 2, swap(2,3) fix 1 — exactly 4 functions.

The function f satisfies f(1) = 1 and f(n+1) = f(n) + 2n for all positive integers n. What is f(6)?

Answer: 31

f(2)=3, f(3)=7, f(4)=13, f(5)=21, f(6)=31; equivalently f(n) = n² − n + 1.

Let g(x) = x² − 3 and f(x) = 2x + 1. For how many integer values of x does f(g(x)) = g(f(x))?

Answer: 0

f(g(x)) = 2x²−5 and g(f(x)) = 4x²+4x−2; setting equal gives 2x²+4x+3 = 0, which has discriminant 16−24 < 0, so no real solutions.

A function f defined for positive reals satisfies f(x/y) = f(x) − f(y) for all x, y > 0, and f(4) = 2. What is f(64)?

Answer: 6

The rule implies f is logarithmic: f(4^k) = 2k, so f(64) = f(4³) = 3 × 2 = 6.